The conditional stability of steady motions of a mechanical isolated system consisting of a rotating carrier body, a material point creating its static imbalance, and a passive automatic balancer consisting of two identical mathematical pendulums are investigated. The pendulums are mounted on the longitudinal axis of the supporting body and move in the plane of static imbalance. The relative motion of the pendulums is impeded by viscous drag forces. It is established that in the case when the imbalance is and the pendulums can eliminate it with a certain margin, there is one basic movement; in the absence of imbalance, there is a one–parameter family of basic movements; in the case of maximum unbalance, which pendulums can eliminate, there is one basic movement, but it generates a pseudo–family of basic motions. It is also established that conditionally asymptotically stable are separate basic motions, if they are isolated, or a family, or pseudo–families of basic motions. For pendulum, ball and liquid automatic balancers, an approximate law is obtained for the variation of large nutation angles in the case of an axisymmetric and non–axisymmetric supporting body. It is established that the rate of change of the nutation angle is significantly affected by the ratio between the axial moments of inertia of the rotating carrier and the coefficient of viscous drag. An empirical formula is proposed for estimating the residual nutation angle, which occurs when the passive automatic balancers (nutation dampers) are incorrectly installed on the spacecraft and stabilized by rotation, and an example of its application for a particular satellite is given. It is shown that incorrect installation of the auto balance on the supporting body can lead to the formation of a residual nutation angle even in the case of a "stable" supporting body. The obtained results can be used in the design of passive automatic balancers (nutation damping) (pendulum, ball and liquid) for spacecrafts stabilized by rotation